Steklov Spectra and Div-curl Analysis

这个项目的中心是分析方程的div-curl系统，并在三维区域中解决具有物理边界条件的偏微分方程。这样的系统作为连续介质力学和电磁场理论中广泛问题的数学模型而出现。特别注意将支付给描述可解性条件和这些系统的弱解的近似和表示。一些新奇涉及到这些问题的标量和矢量势的仔细选择和使用斯捷克洛夫特征函数来描述边界数据的影响。另一类问题是发展的变分原理的特点解决方案的初始值问题的发展方程。在此奖项下进行的分析应有助于发展计算模拟和数值近似的解决方案，许多重要的方程在科学和工程中出现。

This project centers on the analysis of div-curl systems of equations and on the solution of partial differential equations with physical boundary conditions in 3 dimensional regions. Such systems arise as mathematical models of wide classes of problems in continuum mechanics and electromagnetic field theories. Particular attention will be paid to describing solvability conditions and the approximation and representation of weak solutions of these systems. Some of the novelty involves the careful choice of scalar and vector potentials for these problems and the use of Steklov eigenfunctions to describe the effects of boundary data. Another class of problems is the development of variational principles that characterize solutions of initial value problems for evolution equations. The analysis performed under this award should help in the development of computational simulations and the numerical approximation of solutions to many important equations arising in science and engineering.